Do isomorphic groups have the same number of Sylow $p$-subgroups?

Question

I think that isomorphic groups should have the same number of Sylow $p$-groups, but I am not sure why, I am a little stuck on this, I really don't know where to even begin, or if this is even true (sorry I am very new to group theory). I was not able to find anything online. Any help/thoughts?

Thank you!

Answer 1

Yes.

For any isomorphism $\varphi:G\to H$ of groups, we have $K\le G$ if and only if $\varphi(K)\le H$, where $\varphi(K)=\{\varphi(k)\mid k\in K\}$ is the $\varphi$-image of $K$. Therefore, $\varphi$ is a one-to-one correspondence between the subgroups of $G$ with those of $H$.

Moreover, $|G|=|H|$. Isomorphisms preserve the orders of subgroups too, which is to say that $|K|=|\varphi(K)|$ for all $K\le G$.

Can you conclude from here?

Hint: Show that the restriction $$\begin{align}\varphi|_K: K&\to \varphi(K),\\ k&\mapsto \varphi(k)\end{align}$$ of $\varphi$ to $K$ is an isomorphism of $K$ and $\varphi(K)$